Question

a) Find the mean of the Gumbel distribution

b) Derive the mean and variance of the Weibull distribution.

Answer 1

a) Find the mean of the Gumbel distribution

Gumbel Distribution represents the distribution of extreme values either maximum or minimum of samples used in various distributions. It is used to model distribution of peak levels.

The Gumbel distribution with location parameter and scale parameter is

probability density function and distribution function

Probability density function of Gumbel distribution is given as:

$=$

Where −

• α = location parameter.

• β = scale parameter.

• x = random variable.

Cumulative distribution function of Gumbel distribution is given as:

$=$

1.4.) 140-142-1

mean of the Gumbel distribution is

μ = α – βή

where is the Euler-Mascheroni constant

b) Derive the mean and variance of the Weibull distribution.

Dale: Let x denotes the Weibull distribution and the podof of the Weibull distribution is given fixa, B) : & B x - e- xx xxo o ethe where where ayo, Byo The mean of the Weibull distibution. is given by , Toxfx: & iB) da 61X) - = (xapx -ax B -ax 47 ß et let ax =2 then x- Now, differentiate on both sides then. we get, da= a B 1 B da B so, the limits are given by it x=0 then z=0 if x=to then 2=0 -TIPI to Ex) = ß rozez à Z da B --VB 1 ->, YB +1)-dz since the above Gunction form oax integral is a gamma 10-ak nol dx dx-an

so in the above case in place of a=1 and not Hence, the mean of Weibull disph f(x) - 2 Br(it is Variance is Vaz(x) = f(x²) - [E(X)] ² E (X²) - po В B-1 at de 2" е - 2247 z*-! dz VCX **** [C1* %*) - Cerca ***[[(+2)+(1++))